How to wait on a backgrounded sub-process with `wait` command

Question

I have two scripts

foox.sh

#!/bin/bash

echo "foox"
./fooy.sh &
pid=${!}
echo "waiting on ${pid}"
timeout 4 wait ${pid}
echo "waited on ${pid} with result ${!}"

and fooy.sh

#!/bin/bash

echo "fooy.sh"
sleep 99999999

The objective is for foox.sh to wait on fooy.sh for 4 seconds then time out with a non zero exit code on the timeout command.

However, I get that the PID of the fooy.sh process in a not a child process of foox.sh which makes no sense to me, and thus the question.

Here is standard output:

foox
waiting on 48145
fooy.sh
/usr/bin/wait: line 4: wait: pid 48145 is not a child of this shell
waited on 48145 with result 48145